Chapter 2: Boolean Algebra and Logic Gates

1. Chapter 2: Boolean Algebra and Logic Gates

3. F1= XY’ + X’Z F1

4. F2= X’Y’Z + X’YZ + XY’

5. F1 F1= XY’ + X’Z F2= X’Y’Z + X’YZ + XY’

6. F(x, y, z) = xy + x’z + yz = xy + x’z + yz.1 By Postulate 5(a) By Postulate 4(a) By Theorem 2

7. CANONICAL FORMS • How to express a boolean function algebraically from a given truth table? F(x,y,z)=?

8. Summation of Minterms • F(x,y,z)=?

9. Summation of Minterms • F(x,y,z)= m1+ m3 + m4 + m5 = X’Y’Z+X’YZ+XY’Z’+XY’Z =∑(1, 3, 4, 5)

10. Summation of Minterms • F(x,y,z)= m1+ m3 + m4 + m5 = X’Y’Z+X’YZ+XY’Z’+XY’Z =∑(1, 3, 4, 5) • F’ = m0 + m2 + m6 + m7 = X’Y’Z’+X’YZ’+XYZ’+XYZ

11. Multiplication of Maxterms • F(x,y,z)=?

12. Multiplication of Maxterms By MinTerms, F = m1+ m3 + m4 + m5 = X’Y’Z+X’YZ+XY’Z’+XY’Z F’ = m0 + m2 + m6 + m7 = X’Y’Z’+X’YZ’+XYZ’+XYZ . => F = (X’Y’Z’+X’YZ’+XYZ’+XYZ)’ =(X+Y+Z) (X+Y’+Z) (X’+Y’+Z) (X’+Y’+Z’)

13. Multiplication of Max terms By MinTerms, F = m1+ m3 + m4 + m5 = X’Y’Z+X’YZ+XY’Z’+XY’Z F’ = m0 + m2 + m6 + m7 = X’Y’Z’+X’YZ’+XYZ’+XYZ . => F = (X’Y’Z’+X’YZ’+XYZ’+XYZ)’ = (X+Y+Z) (X+Y’+Z) (X’+Y’+Z) (X’+Y’+Z’)= M0 + M2 + M6 + M7 = ∏ ( 0, 2, 6, 7)

14. Conversion Between Canonical Forms Boolean functions expressed as a sum of mintermsor product of maxterms are called canonical form By MinTerms, F = m1+ m3 + m4 + m5 = ∑(1, 3, 4, 5) (function gives 1) By MaxTerms, F = M0 M2 M6 M7 = ∏ ( 0, 2, 6, 7) (function gives 0)

15. Function Expression in Canonical Form Express the Boolean function F = A + B’C as a sum of minterms (All variables must be present in each term) Process: 1 • In each term, for each missing variable P, multiply (P+P’), until all the variables appear in each term • Finally, if same term appears multiple times, discard multiple copies F = A + B’C = A(B+B’) + B’C(A+A’) = AB + AB’ + B’CA + B’CA’ = AB(C+C’) + AB’(C+C’) + B’CA + B’CA’ = ABC + ABC’ + AB’C + AB’C’ + AB’C + A’B’C = ABC + ABC’ + AB’C + AB’C’ + A’B’C = m7 + m6 + m5 + m4 + m1

16. Conversion to Canonical Form Express the Boolean function F = A + B’C as a sum of minterms. Process: 2 • Draw the Truth Table • Read the minterms from the truth table F = m1 + m4 + m5 + m6 + m7

17. Conversion to Canonical Form Express the Boolean function F = xy + x’z as a product of maxterms Process: 1 • Convert the function into OR terms using the distributive law F = xy + x’z = (xy + x’)(xy + z) = (x + x’)(y + x’) (x + z)(y + z) = (x’ + y)(x + z)(y + z) • For each missing variable P, add the PP’ with each term, until all the variables appear in each term. F =(x’ + y)(x + z)(y + z) = (x’ + y + zz’) (x + z + yy’) (y + z + xx’) = (x’ + y + z)(x’ + y + z’) (x + y + z)(x + y’ + z) (x + y + z)(x’ + y + z) = (x’ + y + z)(x’ + y + z’) (x + y + z)(x + y’ + z) Distributive Law: X(Y+Z) = XY + XZ X+(YZ) = (X+Y) (X+Z)

18. Conversion to Canonical Form Express the Boolean function F = xy + x’z as a product of maxterms Process: 2 • Draw the Truth Table • Read the maxterms from the truth table F = M0M2M4M5

19. Standard Form • Each term in the function may contain one, two, or any number of literalsF = y + xy + xyz • Either the sum of products or the products of sums, not both! F =AB + C(D + E)  Non Standard Form = AB + CD + CE  Standard Form Some Standard Functions: F1 = y’ + xy + x’yz’ (sum of products) F2 = x(y’ + z)(x’ + y + z’) (product of sums)

20. Common Logical operations

21. Common Logical operations